two same objects rotating differently
I've fixed it for you.



You needed to keep the Y-axes of the steering wheel aligned to the direction perpendicular to the plane of rotation (shown below). The purpose of parenting the steering wheel to the empty was so the axis of the steering wheel stayed aligned for your desired rotation axis. 



All I needed to do was to enable Transform Only > Origins, and Face Snapping with Align Rotation to Target then snap the origin onto that flat panel in the middle of the steering wheel. This allowed me to realign the origin of the steering wheel with the normals of that surface. However, this places the Z-axis in the direction you wanted the Y-Axis, so I just rotated it 90 degrees about the X-axis.

Then, just to make sure I didn't move the origin, I pressed Shift+S, Selection to cursor. (The 3D cursor was conveniently already at the origin's original location)

 

Edit: Alternative metohd for pre 2.81

Rotate the empty box back to 0 degrees for all axes. This will bring the steering wheel back into an orientation where the global Y-axis matches up with the desired rotational axis. Unparent the steering wheel with Alt+P, and press Ctrl+A and Apply rotation to the steering wheel. Then you can reparent the wheel to the empty and it should work as intended.